Senate Republicans initiated their first attempt of the new Congress to roll back the Affordable Care Act, one of President-elect Donald Trump's biggest campaign promises.

On Tuesday, Sen. Mike Enzi introduced a resolution to repeal Obamacare through the budget reconciliation process, which would allow Republicans to repeal the law with a simple 50-vote majority in the Senate.

"Americans face skyrocketing premiums and soaring deductibles," Enzi said in a statement. "Insurers are withdrawing from markets across the country, leaving many families with fewer choices and less access to care than they had before — the opposite of what the law promised."

Though Obamacare has provided insurance for close to 20 million Americans, Enzi's resolution would initiate a partial repeal of the law without offering a permanent replacement plan, which some congressional Republicans said they would announce in the coming months.

For their part, Democrats plan to resist or make Republicans pay a heavy political price for the repeal of the law, President Barack Obama's signature legislative achievement.

Obama himself is set to visit Congress this week to rally support for the healthcare law. Vice President-elect Mike Pence will meet with House Republicans the same day to counter the president's visit, urging Republicans to pass their own version of healthcare reform.

During a Tuesday interview on MSNBC's "Morning Joe," incoming Trump counselor Kellyanne Conway pledged to craft a plan that allowed the millions of people insured under Obamacare to keep their insurance. She did not specify how any existing Republican plans would allow everyone who is covered through Obamacare to keep their insurance, but referenced familiar Republican proposals, including allowing people to purchase health insurance across state lines and promising to block-grant Medicaid.

"We don't want anyone who currently has insurance to not have insurance," Conway said.